Little q-Jacobi

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Little q-Jacobi

Basic hypergeometric representation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \littleqJacobi{n}@{x}{a}{b}{q}=\qHyperrphis{2}{1}@@{q^{-n},abq^{n+1}}{aq}{q}{qx} }}

Orthogonality relation(s)

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \sum_{k=0}^{\infty}\frac{\qPochhammer{bq}{q}{k}}{\qPochhammer{q}{q}{k}}(aq)^k\littleqJacobi{m}@{q^k}{a}{b}{q}\littleqJacobi{n}@{q^k}{a}{b}{q} {}=\frac{\qPochhammer{abq^2}{q}{\infty}}{\qPochhammer{aq}{q}{\infty}}\frac{(1-abq)(aq)^n}{(1-abq^{2n+1})} \frac{\qPochhammer{q,bq}{q}{n}}{\qPochhammer{aq,abq}{q}{n}}\,\Kronecker{m}{n} }}

Recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle -x\littleqJacobi{n}@{x}{a}{b}{q}=A_n\littleqJacobi{n+1}@{x}{a}{b}{q}-\left(A_n+C_n\right)\littleqJacobi{n}@{x}{a}{b}{q} {}+C_n\littleqJacobi{n-1}@{x}{a}{b}{q} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle C_n=aq^n\frac{(1-q^n)(1-bq^n)}{(1-abq^{2n})(1-abq^{2n+1})}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle A_n=q^n\frac{(1-aq^{n+1})(1-abq^{n+1})}{(1-abq^{2n+1})(1-abq^{2n+2})}}}


Monic recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\moniclittleqJacobi{n}@@{x}{a}{b}{q}=\moniclittleqJacobi{n+1}@@{x}{a}{b}{q}+(A_n+C_n)\moniclittleqJacobi{n}@@{x}{a}{b}{q}+A_{n-1}C_n\moniclittleqJacobi{n-1}@@{x}{a}{b}{q} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle C_n=aq^n\frac{(1-q^n)(1-bq^n)}{(1-abq^{2n})(1-abq^{2n+1})}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle A_n=q^n\frac{(1-aq^{n+1})(1-abq^{n+1})}{(1-abq^{2n+1})(1-abq^{2n+2})}}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \littleqJacobi{n}@{x}{a}{b}{q}=\frac{(-1)^nq^{-\binomial{n}{2}}\qPochhammer{abq^{n+1}}{q}{n}}{\qPochhammer{aq}{q}{n}}\moniclittleqJacobi{n}@@{x}{a}{b}{q} }}

q-Difference equation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle q^{-n}(1-q^n)(1-abq^{n+1})xy(x) {}=B(x)y(qx)-\left[B(x)+D(x)\right]y(x)+D(x)y(q^{-1}x) }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle D(x)=x-1}} &

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle B(x)=a(bqx-1)}} &

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle y(x)=\littleqJacobi{n}@{x}{a}{b}{q}}}


Forward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \littleqJacobi{n}@{x}{a}{b}{q}-\littleqJacobi{n}@{qx}{a}{b}{q} {}=-\frac{q^{-n+1}(1-q^n)(1-abq^{n+1})}{(1-aq)}x\littleqJacobi{n-1}@{x}{aq}{bq}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qderiv{q}\littleqJacobi{n}@{x}{a}{b}{q}=-\frac{q^{-n+1}(1-q^n)(1-abq^{n+1})} {(1-q)(1-aq)}\littleqJacobi{n-1}@{x}{aq}{bq}{q} }}

Backward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle a(bx-1)\littleqJacobi{n}@{x}{a}{b}{q}-(x-1)\littleqJacobi{n}@{q^{-1}x}{a}{b}{q} {}=(1-a)\littleqJacobi{n+1}@{x}{aq^{-1}}{bq^{-1}}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mathcal{D}_{q^{-1}}\left[w(x;\alpha,\beta|q)\littleqJacobi{n}@{x}{q^{\alpha}}{q^{\beta}}{q}\right] {}=\frac{1-q^{\alpha}}{q^{\alpha-1}(1-q)}w(x;\alpha-1,\beta-1|q)\littleqJacobi{n+1}@{x}{q^{\alpha-1}}{q^{\beta-1}}{q} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x;\alpha,\beta|q)=\frac{\qPochhammer{qx}{q}{\infty}}{\qPochhammer{q^{\beta+1}x}{q}{\infty}}x^{\alpha}}}


Rodrigues-type formula

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x;\alpha,\beta|q)\littleqJacobi{n}@{x}{q^{\alpha}}{q^{\beta}}{q} {}=\frac{q^{n\alpha+\binomial{n}{2}}(1-q)^n}{\qPochhammer{q^{\alpha+1}}{q}{n}} \left(\mathcal{D}_{q^{-1}}\right)^n\left[w(x;\alpha+n,\beta+n|q)\right] }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x;\alpha,\beta|q)=\frac{\qPochhammer{qx}{q}{\infty}}{\qPochhammer{q^{\beta+1}x}{q}{\infty}}x^{\alpha}}}


Generating function

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qHyperrphis{0}{1}@@{-}{aq}{q}{aqxt}\,\qHyperrphis{2}{1}@@{x^{-1},0}{bq}{q}{xt}=\sum_{n=0}^{\infty} \frac{(-1)^nq^{\binomial{n}{2}}}{\qPochhammer{bq,q}{q}{n}}\littleqJacobi{n}@{x}{a}{b}{q}t^n }}

Limit relations

Big q-Jacobi polynomial to Little q-Jacobi polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{c\rightarrow -\infty}\bigqJacobi{n}@{cqx}{a}{b}{c}{q}=\littleqJacobi{n}@{x}{a}{b}{q} }}

q-Hahn polynomial to Little q-Jacobi polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim _{N\rightarrow\infty} \qHahn{n}@{q^{x-N}}{\alpha}{\beta}{N}{q}=\littleqJacobi{n}@{q^x}{\alpha}{\beta}{q} }}

Little q-Jacobi polynomial to Little q-Laguerre / Wall polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \littleqJacobi{n}@{x}{a}{0}{q}=\littleqLaguerre{n}@{x}{a}{q} }}

Little q-Jacobi polynomial to q-Laguerre polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{b\rightarrow -\infty}\littleqJacobi{n}@{-b^{-1}q^{-1}x}{q^{\alpha}}{b}{q}= \frac{\qPochhammer{q}{q}{n}}{\qPochhammer{q^{\alpha+1}}{q}{n}}\qLaguerre[\alpha]{n}@{x}{q} }}

Little q-Jacobi polynomial to q-Bessel polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{a\rightarrow 0}\littleqJacobi{n}@{x}{a}{-a^{-1}q^{-1}b}{q}=\qBesselPoly{n}@{x}{b}{q} }}

Little q-Jacobi polynomial to Jacobi / Laguerre polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{q\rightarrow 1}\littleqJacobi{n}@{x}{q^{\alpha}}{q^{\beta}}{q}=\frac{\Jacobi{\alpha}{\beta}{n}@{1-2x}} {\Jacobi{\alpha}{\beta}{n}@{1}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{q\rightarrow 1}\littleqJacobi{n}@{\textstyle\frac{1}{2}(1-q)x}{q^{\alpha}}{-q^{\beta}}{q} =\frac{\Laguerre[\alpha]{n}@{x}}{\Laguerre[\alpha]{n}@{0}} }}

Remarks

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \littleqJacobi{n}@{x}{a}{b}{q}=\frac{\qPochhammer{bq}{q}{n}}{\qPochhammer{aq}{q}{n}}(-1)^nb^{-n}q^{-n-\binomial{n}{2}} \bigqJacobi{n}@{bqx}{b}{a}{0}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qMeixner{n}@{q^{-x}}{b}{c}{q}=\littleqJacobi{n}@{-c^{-1}q^n}{b}{b^{-1}q^{-n-x-1}}{q} }}

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